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arXiv:quant-ph/9708011 · Journal of Physics A Mathematical and General

Continuous stochastic Schrödinger equations and localization

M Rigo, F Mota-Furtado, Patrick F. O’Mahony

Published Nov 7, 1997• 26 citations
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Abstract

The set of continuous norm-preserving stochastic Schrödinger equations associated with the Lindblad master equation is introduced. This set is used to describe the localization properties of the state vector toward eigenstates of the environment operator. Particular focus is placed on determining the stochastic equation which exhibits the highest rate of localization for wide open systems. An equation having such a property is proposed in the case of a single non-Hermitian environment operator. This result is relevant to numerical simulations of quantum trajectories where localization properties are used to reduce the number of basis states needed to represent the system state, and thereby increase the speed of calculation.

DOI
10.1088/0305-4470/30/21/026

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The set of continuous norm-preserving stochastic Schrödinger equations associated with the Lindblad master equation is introduced. This set is used to describe the localization properties of the state vector toward eigenstates of the environment operator. Particular focus is placed on determining the stochastic equation which exhibits the highest rate of localization for wide open systems. An equation having such a property is proposed in the case of a single non-Hermitian environment operator. This result is relevant to numerical simulations of quantum trajectories where localization properties are used to reduce the number of basis states needed to represent the system state, and thereby increase the speed of calculation.
The set of continuous norm-preserving stochastic Schrodinger equations associated with the Lindblad master equation is introduced. This set is used to describe the localization properties of the state vector toward eigenstates of the environment operator. Particular focus is placed on determining the stochastic equation which exhibits the highest rate of localization for wide open systems. An equation having such a property is proposed in the case of a single non-hermitian environment operator. This result is relevant to numerical simulations of quantum trajectories where localization properties are used to reduce the number of basis states needed to represent the system state, and thereby increase the speed of calculation.
We present the stochastic Schr\"odinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it to the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our powerful stochastic approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger as $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$.
Positivity preservation is naturally guaranteed in exact non-Markovian master equations for open quantum system dynamics. However, in many approximated non-Markovian master equations, the positivity of the reduced density matrix is not guaranteed. In this paper we provide a general class of time-local, perturbative, and positivity-preserving non-Markovian master equations generated from stochastic Schr\"odinger equations, particularly quantum-state-diffusion equations. Our method has an expanded range of applicability for accommodating a variety of non-Markovian environments. We show the positivity-preserving master equation for a three-level system coupled to a dissipative bosonic environment as a particular example to exemplify our general approach. We illustrate the numerical simulations with an analysis explaining why the previous approximated non-Markovian master equations cannot guarantee positivity. Our work provides a consistent master equation for studying the non-Markovian dynamics in ultrafast quantum processes and strong-coupling systems.
Weak measurement of a subset of noncommuting observables of a quantum system can be modeled by the open-system evolution, governed by the master equation in the Lindblad form.The open-system density operator can be represented as a statistical mixture over non-unitarily evolving pure states, driven by the stochastic nonlinear Schrödinger equation (sNLSE).The globally stable solution of the sNLSE is obtained in the case where the measured subset of observables comprises the spectrum-generating algebra of the system.This solution is a generalized coherent state (GCS), associated with the algebra.The result is based on proving that the GCS minimizes the trace-norm of the covariance matrix, associated with the spectrum-generating algebra.
A number of authors have proposed stochastic versions of the Schrödinger equation, either as effective evolution equations for open quantum systems or as alternative theories with an intrinsic collapse mechanism. Here we discuss two directions for the generalization of these equations. First, we study a general class of norm preserving stochastic evolution equations, and show that even after making several specializations there is an infinity of possible stochastic Schrödinger equations for which state vector collapse is provable. Second, we explore the problem of formulating a relativistic stochastic Schrödinger equation, using a manifestly covariant equation for a quantum field system based on the interaction picture of Tomonaga and Schwinger. The stochastic noise term in this equation can couple to any local scalar density that commutes with the interaction energy density, and leads to collapse onto spatially localized eigenstates. However, as found in a similar model by Pearle, the equation predicts an infinite rate of energy nonconservation proportional to δ 3 ( ), arising from the local double commutator in the drift term.
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.